Combinatorial constructions of fault-tolerant routings with levelled minimum optical indices

The design of fault-tolerant routings with levelled minimum optical indices plays an important role in the context of optical networks. However, not much is known about the existence of optimal routings with levelled minimum optical indices besides the results established by Dinitz, Ling and Stinson via the partitionable Steiner quadruple systems approach. In this paper, we introduce a new concept of a large set of even levelled P3⃗-design of order vv and index 2, denoted by (v,P3⃗,2)-LELD, which is equivalent to an optimal, levelled (v−2)(v−2)-fault-tolerant routing with levelled minimum optical indices of the complete network with vv nodes. On the basis of the theory of three-wise balanced designs and partitionable candelabra systems, several infinite classes of (v,P3⃗,2)-LELDs are constructed. As a consequence, the existence problem for optimal routings with levelled minimum optical indices is solved for nearly a third of the cases.